LA VITA (1898-1972)
Maurits Cornelis Escher was born on June 17 1898 in Leeuwarden. In Arnheim's high school he had an excellent design master, F.W. VAN DER HAAGEN, who taught him the technique of engraving on linoleum helped develop the predisposition of Escher for graphic art. From 1919 to 1922 he attended the School of Architecture and Decorative Art of Haarlem, where he followed the courses of free graphic techniques of S. JESSURUN DE MESQUITA, whose strong personality greatly influenced his future development as a graphic artist. In 1922 he moved to Italy and settled in Rome in 1924. During the ten years spent in Italy he made numerous study trips. In 1934 he moved to Switzerland for two years and then five to Brussels. From 1941 he lived in Baarn in Holland, where he died on 27 March 1972 at the age of 73.
The OPERA
The work of Escher consists of a series of prints that mainly have Mediterranean landscapes and Southern Italy, produced almost all before 1937, and of 70 more works (after 1937) of nick.
In mathematical background works we can distinguish three themes: the structure of space, the structure of the plane, the relationship between space and surface.
The structure of space
If you look at the prints of Escher in his totality you notice that he does not search the pictorial side as well as, rather, the structure. Despite his ten-year stay in Rome, at the center of the monumental remains of antiquity, Escher will give them just a few pictures and even visits to Pompeii have left no trace in his work. After 1937 he will no longer take care of space structures. The attention to strictly mathematical figures and shapes will develop only later and will merge on the admiration of crystal structures.
Within the space-structure theme there are three categories:
a) landscapes
b) interconnection of different worlds
c) abstract mathematical solids.
The structure of the floor
The development of this theme begins with interest in the regular division of the plan, induced in Escher in particular by visits to Alhambra. After an intense study, which cost him a lot of effort, not being Escher a mathematician, he elaborated a complex system for the regular subdivision of the plan, a system that would later arouse the wonder of crystalographers and mathematicians. In the works dedicated to cycles, in which initial and final phase are interchangeable, the regular division of the plan is normally used. Finally, the regular division of the plan resumed in its infinity approximations.
The structure of the plan is based on three groups of paintings:
a) metamorphosis
b) cycles
c) Infinity approximations.
Relationship between space and surface
There is a well-known problem: how to represent the three dimensions in a two-dimensional surface. On this point Escher submits the laws of perspective, which from the Renaissance apply to spatial representation, to a critical evaluation, of research, and find new laws that illustrate in his perspective paintings. We find three groups of compositions:
a) the essence of representation (conflict-surface space)
(b) perspective
c) impossible figures.
CALEIDES
Kaló (bel) + eîdos (figure) + k)klos (ring)
A kaleidocycle is a three-dimensional ring formed by tetrahedra. The most beautiful and impressive visual achievements of the ring concept are, in my opinion, in the works of Escher. He created some drawings that are among the most conceptually stimulating of all time. Many have their inspiration in paradoxes, illusions or double senses. The mathematicians were among the first admirers of the drawings of Escher, and one understands why: they are often based on mathematical principles of symmetry or regularity. But in a typically escherian design there is much more than simple symmetries and regularity: often there is a basic idea that is made in artistic form. In particular, the ring is one of the most frequent themes in the work of Escher. The concept of rings contains that of infinity: a ring, in fact, is not really a way to represent an endless process in a finite way? In fact infinity intervenes widely in many drawings of Escher. There are often more copies of the same juxtaposition each other in a harmonious way; in the famous Metamorphosis press we see a series of such forms. In Metamorphosis or other paintings, there is already an infinity flood. But in other drawings of Escher appear more disturbing visions of infinity. In some of his designs, a single theme will be able to resume at different levels of reality. For example, a level of design will clearly represent imagination or imagination; another level will represent reality. But what happens if the level chain is not linear but forms a ring? What will then be reality, what will be imagination? The genius of Escher lies in his ability to devise and at the same time to figuratively realize dozens of semi-real and semi-imaginary worlds, worlds full of rings, in which he seems to invite his spectators to enter.
In the volume “Gödel, Escher, Bach: an Eterna Ghirlanda Brillante” by D.R. Hofstadter, the author not only deepens the theme of the ring, of the strange ring, but comes to the conclusion that Escher, in the field of art, is the maximum human approximation conceivable to the idea of recurring and surprisingly captures in some of his images the spirit of the Theorem of Gödel. It is not possible to synthesize, due to banalization, the over 800 series pages and engaging author’s considerations, so we invite the most curious to dedicate a few weeks of their time to this wonderful book.
But let us return to simpler and more practical considerations.
To build a kaleidocycle you start from different identical tetrahedrals, connect, to two to two, obtaining a chain and then join thus creating a closed circle.
One can mathematically demonstrate that there is an infinitely large class of three-dimensional forms. The kaleidocycles are exceptionally symmetrical and the numerous exterior surfaces invite to color them. The coloring of the surfaces causes many geometric effects. If we apply the drawings of Escher to the kaleidocycles, many of his works acquire greater emphasis. To better understand, however, we have to deepen a bit the cyclic division of the plan.
Plan cyclical division
Let's imagine we want to tile a floor. Many identical tiles are placed as in a puzzle to cover the floor evenly. You can put the tiles more or less casual but, usually, there is a reason that is repeated at regular intervals. Such a reason is defined as the cyclical division of the plan. This type of division was the main theme of Escher, indeed his “sperous obsession”. Until the shape of the tiles is an equilateral triangle, or a square or a regular hexagon, it is very easy to find a reason. However, Escher had proposed to devise a cyclical division of the plan, in which the elements represented recognizable figures in nature. Mathematics and crystalgraphs had already theoretically examined all these reasons, finding valid rules for each cyclical division of the plane. After learning about this, Escher interrupted his disappointing experiments and used all his energy and creative talent to develop figures, which overlapped according to his desire. He completed more than 150 color sketches of cyclic patterns, formed by fantastic figures. In the Rettili design a figure of this type escapes from the surface of the drawing block to introduce itself again into its paved world.
The exact definition of a cyclical division of the plan requires that the reason can be reproduced on itself through a fixed shift of direction and path. In mathematical language, this process is called translation of a quantity of points in the plan. All repetitions of any point form a lattice. The order of this lattice does not depend on the choice of the point. The parallel links of the lattice points always give rise to a parallel network. A network of triangles is formed through further parallel divisions. Each cyclical division of the plan is based on such a network of lines: the shape and size of the elements originating in the plan have no importance in this case. The same networks so originated from parallelograms or triangles form, in turn, a paved surface consisting of single pieces equal to each other.
The possible symmetric movements are three: rotation, when the sample rotates around the fixed point; reflection when, drawing a straight line through the sample, a half of the sample becomes the reflected image of the other. Finally, push-up reflection when moving the pattern along a path and then reflect it along the corresponding straight lines.
In his color drawings Escher distributed the colors in such a way that, from time to time, close figures could have different colors becoming a precursor of what is now called “color symmetry”.
Covering a plan without leaving free spaces, using reasons that only intersect through copies is a difficult task. The same Escher has made several attempts to use its periodic motifs, to structure the surface of three-dimensional objects. If you try to decorate the kaleidocycles with the periodic drawings of Escher, it turns out that it is not as easy as it seems. To solve the problem we are helping the “projective geometry”, which is a branch of the geometry that studies what characteristics of an object remain unchanged if it is projected on another surface.
There are different types of projection; the central projection (the one of the slide projector): a point-shaped source emits a bright cone that enlarges the image and reproduces it on a plane. In the parallel projection the image is transmitted through parallel rays. The essential features of the cyclic division of the plane, which is needed for an uninterrupted coating of the kaleidocycles, are maintained only with the parallel projection.
Escher had a strict criterion for the coloring of the cyclic divisions of the plane: two adjacent motifs had to have different colors; only through the contrast of colors, in fact, can be found in a sketch of identical copies individual motifs in a recognizable way. It may seem strange, and yet the problems that arise in coloring fall within the scope of mathematics. Given a certain coloring structure (like a map), the mathematicians ask: how many different colors do you need to solve the problem? How many different ways can I color the structure? Can colours be distributed so that certain colour combinations are necessarily displayed? It is extremely difficult to answer these questions, especially if the structure is complicated and coloring under strict rules. combinatorial calculation, graphic theory and topology are the branches of modern mathematics that deals with these problems.
The problem of the minimum number of colors you need for any map placed on the floor or on a spherical surface, has remained for more than two hundred years unresolved, despite many mathematicians capable have sought a solution. Four colors are enough to color a plan design according to the requirements for maps; for the cyclic divisions of the plane is better if the colors emphasize the symmetries of the design. Before a systematic solution given by mathematicians and crystalographers, Escher had long worked and ordered in systems the possibilities found experimentally.
MODELS OF CALEIDOCILI
Escher possessed a brilliant imagination and was an extremely skillful artifice; however, the reason for the extraordinary effects of his designs is to be sought in mathematics. Not the parallel in the sense of numbers and equations that comes to mind immediately to most of us, but the geometry, the classic and the modern one. Although Escher with his imagination could imagine the fantastic effects he wanted to create graphically, the means necessary for the realization of such effects remained the mathematics. In this regard he read technical treaties and corresponded with mathematicians and crystallographers. In these letters, Escher's modest opinion is evident in regard to his mathematical knowledge. Nevertheless, in his drawings, Escher demonstrates that he has a good mastery of the fundamental principles of mathematics.
The models presented here are the work of Doris Schattschneider, professor of mathematics at Moravian College in Bethlehem, Pensilvania and Fallace Walzer, New York artist, who has supervised drawing preparation and production of copies of the models.
The geometric solids, at the base of the models, are: the tetrahedron and theicosaedron (formed by equilateral triangles), the cube (six squares), the dodecahedron (twelve regular pentagons), the cube-octahedron (six squares and eight equilateral triangles) and the hexagonal kaleidocicles, formed by a grid of triangles.
Models
(the explanatory description is placed alongside the individual models exposed).
RETTILI, tetraedro
Three MONDI, octaedron
FARFALLE, icosahedron
PESCE, cube
CONCHIGLIE AND MARINE, dodecaedro
INCONTRO, kaleidociclo ritorto
COLEOTTERI, hexagonal kaleidocycle
PESCE, hexagonal kaleidocycle
Three ELEMS, hexagonal kaleidocycle
PESCE, ANATRA, SAURO, hexagonal kaleidocycle
VERBUM, hexagonal kaleidocycle
FIORI, square kaleidocycle
PARADISO AND INFERNO, square kaleidocycle
SAURI, square kaleidocycle
STEEL FISHES
21 more solids are presented covered by the best known periodic drawings of Escher, with a very pleasant optical effect.
EXHIBITION PROSEGUE...
To better understand the exceptional production of Escher, on the edge of the exhibition The kaleidocicli of Escher, some multimedia stations have been set up that allow to view, in the smallest details, over 700 works of Escher; to deepen their life and to see Escher at work thanks to vintage films and also to manipulate interactively the images of Escher to create custom designs, to play with perspective, to solve colors.
The exhibition closes with the exhibition of the 76 most significant drawings of Escher, each with explanatory caption, according to the following subdivision:
First works (1-7): seven drawings chosen from numerous works made before 1937.
Regular division of the plan (8-35): the richest section, with reflected images, figures with background, shape and contrasts, the infinity number, illustrated stories, irregular surface filling.
Unlimited space (Unlimited space (36-38): unlimited space, cubic space division, three intersecting floors.
Space circles and spirals (39-46): closed knots, Moebius tapes, spirals.
Reflex images (47-54): in water, spherical.
Inversions (55-56): cube, convex and concave.
Polyhedra (57-62): tetrahedra, dodecahedra, regular solids.
Relativity (63-67): perspective on the horizon, nadir and zenit.
Conflict between surface and space (68-73): three-dimensional space and spatial illusions.
Impossible buildings (74,75,76): the three best known and reproduced designs: Belvedere, Ascent and descent, Waterfall.
Throughout his life Maurits Cornelis Escher confessed to his inability to understand the mathematics, declaring himself “completely fasting studies and knowledge in the field of exact sciences”. However, as a child, he was fascinated by order and symmetry. That attraction led him to study the configurations of the Alhambra tiles in Granata, to examine geometric drawings in articles of mathematics and finally to follow his original ideas on floor tipping. The attention that Escher gave to the colors of his designs of interconnected dowels anticipated the subsequent research of mathematicians and crystalgraphs in the field of color symmetry.
Escher wrote that one of his main thrusts was “a deep interest in the geometric laws contained in the nature that surrounds us” and attempted to give a visual expression to abstract concepts and to represent the ambiguities of observation and human knowledge. By doing so, he often found himself in a world governed by mathematics. Escher was even obsessed with the concept of “ordinary division of the plan” and realized over 150 color drawings that demonstrate his ingeniousness in creating figures that fill the plane of their clones. These drawings illustrate symmetries of many different types, but for Escher the division of the plane was also a means to capture infinity.
In 1957 the mathematician H.S.M. Coxeter sent Escher a reprint of his article in which he illustrated the symmetry of the plan with some drawings of Escher. There the artist found a figure that gave him “a real shock”, a hyperbolic tile of triangles that showed just the desired effect. Four years later, Escher devised his solution to the problem of infinity within a rectangle. The recurring algorithm found by him gives rise to a configuration always similar to itself, in which each element is in relation to another because of a change of scale. Curiosously, self-similar configurations give us examples of figures that have fractional dimensions (or fractals), an ambiguity that Escher would certainly have appreciated.
Towards the end of life, Escher wrote: “Of course, I’m glad that from all of this is Contact Details and friendships with mathematicians. They often gave me new ideas, and sometimes there was even an interaction between us. How can they be playful, these gentlemen and these ladies of great culture!”
Bruno Ernst,The Magic Mirror of M.C. Escher, Evergreen, 1996
M.C. Escher,Exploring Infinity, Garzanti, 1991
R. Hofstadter,Gödel, Escher, Bach: an Eterna Ghirlanda Brillante, Adelphi Editions, 2001
M.C. Escher, Graphics and Designs, Evergreen, 1992
D. Schattschneider, M.C. Escher Caleidocicli, Taschen, 1992
Andrea Bonfiglioli and Camilla Valentini, Mathematics Art and Technology: from Escher to Computer Graphics, Editions Aspasia, 2000
D. Schattschneider,The metaphors of Escher, The Sciences n. 317, January 1995
M. Fellows, The life and works of Escher, Siena, 1995
M.C. Escher, Book of Boxes, Taschen, 1998
Albert Flocon, A’ la frontière de l’art graphique et des mathématiques:Maurits Cornelis Escher, in « Jardin des Arts «,131, 1957
Martin Gardner, The Eerie Mathematical Art of M.C. Escher, in “Scientific American”, vol.214, April 1966
J.L. Locher, The World of Escher, Milan, 1978
Beard Col. R.S., Patterns in Space, Creative Publications,Palo Alto, California,1973
Bool F. H. and others, Escher,His Life and Complete Graphic Work, NY, 1982
Coxeter H.S.M., Introduction to Geometry, NY, 1969
Coxeter H.S.M., M.C. Art and Science, Amsterdam, 1987
Cundy H.N., Mathematical Models, Oxford, 1961
(This online report contains only the text; the graphic part (heavy several dozen megabytes) has been excluded to allow an easy and fast download).